In all early civilizations, the first expression of mathematical understanding appears in the form of counting systems. Numbers in very early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome). Although today, we take our decimal system for granted, not all ancient civilizations based their numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.
The Decimal System in Harappa
In India a decimal system was already in place during the Harappan period, as indicated by an analysis of Harappan weights and measures. Weights corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been identified, as have scales with decimal divisions. A particularly notable characteristic of Harappan weights and measures is their remarkable accuracy. A bronze rod marked in units of 0.367 inches points to the degree of precision demanded in those times. Such scales were particularly important in ensuring proper implementation of town planning rules that required roads of fixed widths to run at right angles to each other, for drains to be constructed of precise measurements, and for homes to be constructed according to specified guidelines. The existence of a gradated system of accurately marked weights points to the development of trade and commerce in Harappan society.
Mathematical Activity in the Vedic Period
In the Vedic period, records of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. However, as in many other early agricultural civilizations, the study of arithmetic and geometry was also impelled by secular considerations. Thus, to some extent early mathematical developments in India mirrored the developments in Egypt, Babylon and China . The system of land grants and agricultural tax assessments required accurate measurement of cultivated areas. As land was redistributed or consolidated, problems of mensuration came up that required solutions. In order to ensure that all cultivators had equivalent amounts of irrigated and non-irrigated lands and tracts of equivalent fertility - individual farmers in a village often had their holdings broken up in several parcels to ensure fairness. Since plots could not all be of the same shape - local administrators were required to convert rectangular plots or triangular plots to squares of equivalent sizes and so on. Tax assessments were based on fixed proportions of annual or seasonal crop incomes, but could be adjusted upwards or downwards based on a variety of factors. This meant that an understanding of geometry and arithmetic was virtually essential for revenue administrators. Mathematics was thus brought into the service of both the secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge (rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC) which describe techniques for the construction of ritual altars in use during the Vedic era. It is likely that these texts tapped geometric knowledge that may have been acquired much earlier, possibly in the Harappan period. Baudhayana's Sutra displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent (or multiple, or fractional) area (such as a square). While some of the formulations are approximations, others are accurate and reveal a certain degree of practical ingenuity as well as some theoretical understanding of basic geometric principles. Modern methods of multiplication and addition probably emerged from the techniques described in the Sulva-Sutras..
Philosophy and Mathematics
Philosophical doctrines also had a profound influence on the development of mathematical concepts and formulations. Like the Upanishadic world view, space and time were considered limitless in Jain cosmology. This led to a deep interest in very large numbers and definitions of infinite numbers. Infinite numbers were created through recursive formulae, as in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).
Jain set theory probably arose in parallel with the Syadvada system of Jain epistemology in which reality was described in terms of pairs of truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates an understanding of the law of indeces and uses it to develop the notion of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with indeterminate equations and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Buddhist mathematics was classified either as Garna (Simple Mathematics) orSankhyan (Higher Mathematics). Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).
The Indian Numeral System
Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal, or near instantaneous motion of the moon, and express it in the form of a basic differential equation. Aryabhatta's equations were elaborated on by Manjula(10th C) and Bhaskaracharya (12th C) who derived the differential of the sine function. Later mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of the Islamic invasions and the conversion of colleges and universities to madrasahs. But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. Although Arab scholars relied on a variety of sources including Babylonian, Syriac, Greek and some Chinese texts, Indian mathematical texts played a particularly important role. Scholars such as Ibn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C, Damascus, author of The book of Chapters in Indian Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba) were amongst the many who based their own scientific texts on translations of Indian treatises. Records of the Indian origin of many proofs, concepts and formulations were obscured in the later centuries, but the enormous contributions of Indian mathematics was generously acknowledged by several important Arabic and Persian scholars, especially in Spain. Abbasid scholar Al-Gaheth wrote: " India is the source of knowledge, thought and insight”. Al-Maoudi (956 AD) who travelled in Western India also wrote about the greatness of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and court historian was amongst the most enthusiastic in his praise of Indian civilization, and specially remarked on Indian achievements in the sciences and in mathematics. Of course, eventually, Indian algebra and trigonometry reached Europe through a cycle of translations, traveling from the Arab world to Spain and Sicily, and eventually penetrating all of Europe. At the same time, Arabic and Persian translations of Greek and Egyptian scientific texts become more readily available in India.
Indians invented the hookah as a way to purify the smoke from tobacco. It became a symbol of nobility.
The Indians were the first people to use large numbers used in complex mathematics and logic. They even had distinct names for these numbers, which went up to 1x10^23^621. They also were the first to use 0 as a actuall number, and grasp the concept of infinty.
Unfortunately, the context of science in South Asian relations has been overwhelmed by competitive defense technologies. While art and music groups are frequently allowed to cross borders between India and Pakistan for performances, scientists have a much more difficult time. In 2007, the U.S. National Science Foundation supported a series of collaborative workshops between Pakistani and Indian environmental scientists, but both countries were resistant to grant visas and the organizers were forced to arrange separate domestic meetings and one joint meeting in Kathmandu, Nepal, where neither side needed a visa. Moreover, the goal of collaborative fieldwork still eludes us.
Even though environmental scientists have little interest in nuclear secrets, the perception of scientists as a security risk remains strong on both sides. The United States could and should play a more active role in building trust between India and Pakistan around nonpolitical issues.
Collaboration on climate change science in the glaciated headwaters of the Indus basin river system, especially following the devastating floods of 2010, makes practical and political sense. It is understandable that India will once again be reluctant to accept any "outside interference" on this but the threat of climate change is a global concern and the Karakoram glaciers are a pivotal natural laboratory for understanding these dynamics. Scientists from Pakistan and India have a clear and present interest in collaborating on this matter as part of their obligations to international environmental agreements.
As delegates assemble for climate change negotiations this coming week in Cancun, Mexico, the instrumental use of science for diplomacy presents an opportunity for unexpected gain amid otherwise sluggish talks. Such deliberations could also encompass a dialogue between scientists on how Earth observation and remote sensing can be effectively used to address environmental security challenges in the future. Currently, there are around ten Indian Remote Sensing (IRS) satellites, which are some of the best in the world for generating information on natural resources. Data from the IRS satellites are used for a variety of applications such as drought monitoring, flood risk zone mapping, urban planning, forestry surveys, environmental impact analysis, and coastal studies.
Development of Philosophical Thought and Scientific Method in Ancient India
Contrary to the popular perception that Indian civilization has been largely concerned with the affairs of the spirit and "after-life", India's historical record suggests that some of the greatest Indian minds were much more concerned with developing philosophical paradigms that were grounded in reality. The premise that Indian philosophy is founded solely on mysticism and renunciation emanates from a colonial and orientalist world view that seeks to obfuscate a rich tradition of scientific thought and analysis in India.
Much of the evidence for how India's ancient logicians and scientists developed their theories lies buried in polemical texts that are not normally thought of as scientific texts. While some of the treatises on mathematics, logic, grammar, and medicine have survived as such - many philosophical texts enunciating a rational and scientific world view can only be constructed from extended references found in philosophical texts and commentaries by Buddhist and Jain monks or Hindu scholars (usually Brahmins).
Although these documents are usually considered to lie within the domain of religious studies, it should be pointed out that many of these are in the form of extended polemics that are quite unlike the holy books of Christianity or Islam. These texts attempt to debate the value of the real-world versus the spiritual-world. They attempt to counter the theories of the atheists and other skeptics. But in their attempts to prove the primacy of a mystical soul or "Atman" - they often go to great lengths in describing competing rationalist and worldly philosophies rooted in a more realistic and more scientific perception of the world. Their extensive commentaries illustrate the popular methods of debate, of developing a hypothesis, of extending and elaborating theory, of furnishing proofs and counter-proofs.
It is also important to note that originally, the Buddhist world view was an essentially atheistic world view. The ancient Jains were agnostics, and within the broad stream of Hinduism - there were several heterodox currents that asserted a predominantly atheistic view. In that sense, these were not religions as we think of today since the modern understanding of religion presumes faith or belief in a super-natural entity.
That so many scholars from each of these philosophical schools felt the imperative to prove their extra-worldly theories using rationalist tools of deductive and inductive logic suggests that faith in a super-natural being could not have been taken for granted. This is borne out by the memoirs of Hieun Tsang (the Chinese chronicler who traveled extensively in India during the 7th C. AD) who describes the merchants of Benaras as being mostly "unbelievers"! He also wrote of intense polemics and debates amongst followers of different Buddhist sects.
Similiarly, there is other evidence that suggests that amongst the intellectuals of ancient India, atheism and skepticism must have been very powerful currents that required repeated and vigorous attempts at persuasion and change. Nevertheless, over centuries, the intellectual discords between the believers and non-believers became more and more muted. The advocates of mystic idealism prevailed over the skeptics, so that eventually, (at the popular level) each of these philosophies functioned as traditional religions with their pantheon of gods and goddesses enticing and lulling most into an intellectual stupor. But at no point were the advocates of "pure faith" ever powerful enough to completely extinguish the rationalist current that had so imbued Indian philosophy.
The Age of Science and Reason
But even amongst those Indian philosophers who accepted the separation of mind and body and argued for the existence of the soul, there was considerable dedication to the scientific method and to developing the principles of deductive and inductive logic. From 1000 B.C to the 4th C A.D (also described as India's rationalistic period) treatises in astronomy, mathematics, logic, medicine and linguistics were produced. The philosophers of the Sankhya school, the Nyaya-Vaisesika schools and early Jain and Buddhist scholars made substantial contributions to the growth of science and learning. Advances in the applied sciences like metallurgy, textile production and dyeing were also made.
In particular, the rational period produced some of the most fascinating series of debates on what constitutes the "scientific method": How does one separate our sensory perceptions from dreams and hallucinations? When does an observation of reality become accepted as fact, and as scientific truth? How should the principles of inductive and deductive logic be developed and applied? How does one evaluate a hypothesis for it's scientific merit? What is a valid inference? What constitutes a scientific proof?
These and other questions were attacked with an unexpected intellectual vigour. As keen observers of nature and the human body, India's early scientist/philosophers studied human sensory organs, analyzed dreams, memory and consciousness. The best of them understood dialectics in nature - they understood change, both in quantitative and qualitative terms - they even posited a proto-type of the modern atomic theory. It was this rational foundation that led to the flowering of Indian civilization.
This is borne out by the testaments of important Greek scientists and philosophers of that period. Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. (The famous Pythagoras theorem is actually a restatement of a result already known and recorded by earlier Indian mathematicians). Later, Herodotus (father of Greek history) was to write that the Indians were the greatest nation of the age. Megasthenes - who travelled extensively through India in the 4th C. B.C also left extensive accounts that paint India in highly favorable light (for that period).
Intellectual contacts between ancient Greece and India were not insignificant. Scientific exchanges between Greece and India were mutually beneficial and helped in the development of the sciences in both nations. By the 6th C. A.D, with the help of ancient Greek and Indian texts, and through their own ingenuity, Indian astronomers made significant discoveries about planetary motion. An Indian astronomer - Aryabhata, was to become the first to describe the earth as a sphere that rotated on it's own axis. He further postulated that it was the earth that rotated around the sun and correctly described how solar and lunar eclipses occurred.
Because astronomy required extremely complicated mathematical equations, ancient Indians also made significant advances in mathematics. Differential equations - the basis of modern calculus were in all likelihood an Indian invention (something essential in modeling planetary motions). Indian mathematicians were also the first to invent the concept of abstract infinite numbers - numbers that can only be represented through abstract mathematical formulations such as infinite series - geometric or arithmetic. They also seemed to be familiar with polynomial equations (again essential in advanced astronomy) and were the inventors of the modern numeral system (referred to as the Arabic numeral system in Europe).
The Decimal System in Harappa
Mathematical Activity in the Vedic Period
Philosophy and Mathematics
The Indian Numeral System
Emergence of Calculus
The Spread of Indian Mathematics
Indians invented the hookah as a way to purify the smoke from tobacco. It became a symbol of nobility.
The Indians were the first people to use large numbers used in complex mathematics and logic. They even had distinct names for these numbers, which went up to 1x10^23^621. They also were the first to use 0 as a actuall number, and grasp the concept of infinty.
http://www.philforhumanity.com/Infinity_Minus_Infinity.html
http://www.buzzle.com/articles/south-asian-culture.html
Development of Philosophical Thought and Scientific Method in Ancient India
The Age of Science and Reason